# Complex series doesnt converge absolutely

The series $\sum_{n = 1}^{\infty} a_{n}$ with $a_{n} = \frac{1}{\sqrt n - i n}$ does not converge absolutely because, $$|a_{n}| = \frac{1}{|\sqrt n - i n|} = \frac{1}{\sqrt{n + n^2}} \geq \frac{1}{\sqrt 2 \ n }$$ and $\frac{1}{\sqrt 2}\sum_{n=1}^{\infty} \frac{1}{n}$ does not converge.

Therefore the ratio test and other comparison tests dont work. My question is, if this series does converge or if it diverges. I dont managed to show if the series is either a cauchy sequence or not

• You are correct. It does not converge absolutely. In fact, it does not converge at all. – GEdgar Jun 24 '18 at 21:02
• Im sure there are converging series that dont converge absolutely – haddel Jun 24 '18 at 21:07
• $\sum\frac{(-1)^n}{n}$ converges, but not absolutely. – GEdgar Jun 24 '18 at 21:09

$$\frac{1}{\sqrt{n}-in} = \frac{\sqrt{n}}{n+n^2}+i\frac{n}{n+n^2}$$ From the definition of a complex series, it doesn't converge, since it's imaginary part doesn't converge. ($\frac{n}{n+n^2}$ is comparable with $\frac{1}{n}$ and $\sum \frac{1}{n}$ doesn't converge)